cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 87 results. Next

A255177 Second differences of seventh powers (A001015).

Original entry on oeis.org

1, 126, 1932, 12138, 47544, 140070, 341796, 730002, 1412208, 2531214, 4270140, 6857466, 10572072, 15748278, 22780884, 32130210, 44327136, 59978142, 79770348, 104476554, 134960280, 172180806, 217198212, 271178418
Offset: 0

Views

Author

Luciano Ancora, Feb 21 2015

Keywords

Examples

			Second differences:  1, 126, 1932, 12138,  47544, ... (this sequence)
First differences:   1, 127, 2060, 14324,  63801, ... (A152726)
----------------------------------------------------------------------
The seventh powers:  1, 128, 2187, 16384,  78125, ... (A001015)
----------------------------------------------------------------------
First partial sums:  1, 129, 2316, 18700,  96825, ... (A000541)
Second partial sums: 1, 130, 2446, 21146, 117971, ... (A250212)
Third partial sums:  1, 131, 2577, 23723, 141694, ... (A254641)
Fourth partial sums: 1, 132, 2709, 26432, 168126, ... (A254646)
Fifth partial sums:  1, 133, 2842, 29274, 197400, ... (A254684)

Links

Crossrefs

Cf. A000541, A001015, A152726, A250212, A254641, A254646, A254684, A255178, A255179.

Programs

  • Magma
    [1] cat [14*(-1+n)*(9-22*n+23*n^2-12*n^3+3*n^4): n in [2..30]]; // Vincenzo Librandi, Mar 12 2015
  • Mathematica
    Join[{1}, Table[14 n (3 n^4 + 5 n^2 + 1), {n, 1, 30}], {n, 0, 24}] (* or *)
CoefficientList[Series[(1 + 120 x + 1191 x^2 + 2416 x^3 + 1191 x^4 + 120 x^5 + x^6)/(1 - x)^6, {x, 0, 22}], x]
  • Python
    def A255177(n): return 14*n*(n**2*(3*n**2 + 5) + 1) if n else 1 # Chai Wah Wu, Oct 07 2024

Formula

G.f.: (1 + 120*x + 1191*x^2 + 2416*x^3 + 1191*x^4 + 120*x^5 + x^6)/(1 - x)^6.
a(n) = 14*n*(3*n^4 + 5*n^2 + 1) for n>0, a(0)=1.
a(n) = A022523(n)-A022523(n-1). - R. J. Mathar, Jul 16 2015

Extensions

Edited by Bruno Berselli, Mar 19 2015

A254646 Fourth partial sums of seventh powers (A001015).

Original entry on oeis.org

1, 132, 2709, 26432, 168126, 804552, 3136014, 10459968, 30856839, 82407052, 202678203, 465069696, 1005729452, 2066218896, 4058958828, 7664805504, 13974953853, 24692818836, 42415687153, 71020845504, 116186669130, 186085891160, 292296070170, 450981236160, 684408934755
Offset: 1

Views

Author

Luciano Ancora, Feb 05 2015

Keywords

Examples

			First differences:   1, 127, 2059, 14197,  61741, ...  (A022523)
----------------------------------------------------------------------
The seventh powers:  1, 128, 2187, 16384,  78125, ...  (A001015)
----------------------------------------------------------------------
First partial sums:  1, 129, 2316, 18700,  96825, ...  (A000541)
Second partial sums: 1, 130, 2446, 21146, 117971, ...  (A250212)
Third partial sums:  1, 131, 2577, 23723, 141694, ...  (A254641)
Fourth partial sums: 1, 132, 2709, 26432, 168126, ...  (this sequence)

Links

Crossrefs

Cf. A000541, A001015, A022523, A250212, A254641, A254645.

Programs

  • GAP
    List([1..30], n-> Binomial(n+4,5)*(3*(n+2)^6 -40*(n+2)^4 +151*(n+2)^2 -108)/198); # G. C. Greubel, Aug 28 2019
  • Magma
    [Binomial(n+4,5)*(3*(n+2)^6 -40*(n+2)^4 +151*(n+2)^2 -108)/198: n in [1..30]]; // G. C. Greubel, Aug 28 2019
  • Maple
    seq(binomial(n+4,5)*(3*(n+2)^6 -40*(n+2)^4 +151*(n+2)^2 -108)/198, n=1..30); # G. C. Greubel, Aug 28 2019
  • Mathematica
    Table[n (1 + n) (2 + n) (3 + n) (4 + n) (48 - 100 n - 89 n^2 + 160 n^3 + 140 n^4 + 36 n^5 + 3 n^6)/23760, {n, 20}] (* or *)
Accumulate[Accumulate[Accumulate[Accumulate[Range[20]^7]]]] (* or *)
CoefficientList[Series[(1 + 120 x + 1191 x^2 + 2416 x^3 + 1191 x^4 + 120 x^5 + x^6)/(- 1 + x)^12, {x, 0, 19}], x]
  • PARI
    a(n)=n*(1+n)*(2+n)*(3+n)*(4+n)*(48-100*n-89*n^2+160*n^3+140*n^4 +36*n^5+3*n^6)/23760 \\ Charles R Greathouse IV, Oct 07 2015
  • Sage
    [binomial(n+4,5)*(3*(n+2)^6 -40*(n+2)^4 +151*(n+2)^2 -108)/198 for n in (1..30)] # G. C. Greubel, Aug 28 2019

Formula

G.f.: x*(1 +120*x +1191*x^2 +2416*x^3 +1191*x^4 +120*x^5 +x^6)/(1-x)^12.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(48 - 100*n - 89*n^2 + 160*n^3 + 140*n^4 + 36*n^5 + 3*n^6)/23760.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + n^7.

A254684 Fifth partial sums of seventh powers (A001015).

Original entry on oeis.org

1, 133, 2842, 29274, 197400, 1001952, 4137966, 14597934, 45454773, 127861825, 330540028, 795609724, 1801339176, 3867558072, 7926516900, 15591322404, 29566276257, 54259095093, 96674782246, 167695627750, 283882296880
Offset: 1

Views

Author

Luciano Ancora, Feb 12 2015

Keywords

Examples

			First differences:   1, 127, 2060, 14324,  63801, ...  (A152726)
----------------------------------------------------------------------
The seventh powers:  1, 128, 2187, 16384,  78125, ...  (A001015)
----------------------------------------------------------------------
First partial sums:  1, 129, 2316, 18700,  96825, ...  (A000541)
Second partial sums: 1, 130, 2446, 21146, 117971, ...  (A250212)
Third partial sums:  1, 131, 2577, 23723, 141694, ...  (A254641)
Fourth partial sums: 1, 132, 2709, 26432, 168126, ...  (A254646)
Fifth partial sums:  1, 133, 2842, 29274, 197400, ...  (this sequence)

Links

Crossrefs

Cf. A000541, A001015, A152726, A250212, A254641, A254646, A254681, A254682, A254683.

Programs

  • Mathematica
    Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (- 3 + 5 n + n^2) (- 2 + 5 n + n^2) (5 + 5 n + n^2)/95040, {n,21}] (* or *)
CoefficientList[Series[(- 1 - 120 x - 1191 x^2 - 2416 x^3 - 1191 x^4 - 120 x^5 - x^6)/(-1 + x)^13, {x,0,20}], x]
  • PARI
    a(n)=n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(-3+5*n+n^2)*(-2+5*n+n^2)*(5+5*n+n^2)/95040 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (- x - 120*x^2 - 1191*x^3 - 2416*x^4 - 1191*x^5 - 120*x^6 - x^7)/(- 1 + x)^13.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(-3 + 5*n + n^2)*(-2 + 5*n + n^2)*(5 + 5*n + n^2)/95040.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + n^7.

A250212 Second partial sums of seventh powers (A001015).

Original entry on oeis.org

1, 130, 2446, 21146, 117971, 494732, 1695036, 4992492, 13072917, 31153342, 68720938, 142120342, 278268263, 519829688, 932250488, 1613106744, 2704301673, 4407716634, 7005003334, 10882290034, 16560665275, 24733398404, 36310956980, 52474986980, 74742532605, 105041888406
Offset: 1

Views

Author

Luciano Ancora, Jan 18 2015

Keywords

Comments

The general formula for the second partial sums of m-th powers is: b(n,m) = (n+1)*F(m) - F(m+1), where F(m) is the m-th Faulhaber’s polynomial.

Links

Crossrefs

Cf. A239094 (same sequence, shifted by 1).

Programs

  • GAP
    List([1..30], n-> Binomial(n+2, 3)*(5*(n+1)^6 -25*(n+1)^4 +38*(n+ 1)^2 -12)/60); # G. C. Greubel, Aug 28 2019
  • Magma
    [(n*(n + 1)*(n + 2)*(5*n^6 + 30*n^5 + 50*n^4 -37*n^2 + 6*n + 6) / 360): n in [1..30]]; // Vincenzo Librandi, Jan 22 2015
  • Maple
    seq(binomial(n+2, 3)*(5*(n+1)^6 -25*(n+1)^4 +38*(n+1)^2 -12)/60, n=1..30); # G. C. Greubel, Aug 28 2019
  • Mathematica
    Accumulate[Accumulate[Range[25]^7]] (* Robert G. Wilson v, Jan 21 2015 *)
Table[(n(n+1)(n+2)(5n^6+30n^5+50n^4-37n^2+6n+6)/360), {n, 30}] (* Vincenzo Librandi, Jan 22 2015 *)
RecurrenceTable[{a[n]==2a[n-1]-a[n-2]+n^7, a[1]==1,a[2]==130}, a, {n,30}] (* Bruno Berselli, Jan 22 2015 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,130,2446,21146,117971,494732,1695036,4992492,13072917,31153342},30] (* Harvey P. Dale, Jan 19 2020 *)
  • PARI
    vector(50, n, n*(n+1)*(n+2)*(5*n^6 + 30*n^5 + 50*n^4 - 37*n^2 + 6*n + 6)/360) \\ Michel Marcus, Jan 21 2015
  • Sage
    [binomial(n+2, 3)*(5*(n+1)^6 -25*(n+1)^4 +38*(n+1)^2 -12)/60 for n in (1..30)] # G. C. Greubel, Aug 28 2019

Formula

a(n) = n*(n+1)*(n+2)*(5*n^6 + 30*n^5 + 50*n^4 - 37*n^2 + 6*n + 6)/360.
a(n) = 2*a(n-1) - a(n-2) + n^7.
G.f.: x*(1 +120*x +1191*x^2 +2416*x^3 +1191*x^4 +120*x^5 +x^6)/(1-x)^10. - Georg Fischer, May 24 2019
a(n) = A239094(n+1). - Danny Rorabaugh, Apr 22 2015

A254641 Third partial sums of seventh powers (A001015).

Original entry on oeis.org

1, 131, 2577, 23723, 141694, 636426, 2331462, 7323954, 20396871, 51550213, 120271151, 262391493, 540659756, 1060489444, 1992739932, 3605846676, 6310148349, 10717864983, 17722868317, 28605158351, 45165823626, 69899222030, 106210179010, 158685165990
Offset: 1

Views

Author

Luciano Ancora, Feb 05 2015

Keywords

Links

Crossrefs

Cf. A254640, A254642, A254643.

Programs

  • GAP
    List([1..30], n-> Binomial(n+3,4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2 -6*n+6)/30); # G. C. Greubel, Aug 28 2019
  • Magma
    [Binomial(n+3,4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2-6*n+6)/30: n in [1..30]]; // G. C. Greubel, Aug 28 2019
  • Maple
    seq(binomial(n+3,4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2-6*n+6)/30, n=1..30); # G. C. Greubel, Aug 28 2019
  • Mathematica
    Table[n(1+n)(2+n)(3+n)(6 -6n -20n^2 +15n^3 +25n^4 +9n^5 +n^6)/720, {n, 30}]
CoefficientList[Series[(1 +120x +1191x^2 +2416x^3 +1191x^4 +120x^5 + x^6)/(1-x)^11, {x, 0, 30}], x]
Nest[Accumulate,Range[30]^7,3] (* or *) LinearRecurrence[{11,-55,165, -330,462,-462,330,-165,55,-11,1},{1,131,2577,23723, 141694, 636426, 2331462, 7323954,20396871,51550213,120271151},30] (* Harvey P. Dale, Jun 19 2018 *)
  • PARI
    Vec((1 +120*x +1191*x^2 +2416*x^3 +1191*x^4 +120*x^5 +x^6)/(1-x)^11 + O(x^40)) \\ Andrew Howroyd, Nov 06 2018
  • PARI
    vector(30, n, binomial(n+3,4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2 -6*n+6)/30) \\ G. C. Greubel, Aug 28 2019
  • Sage
    [binomial(n+3,4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2-6*n+6)/30 for n in (1..30)] # G. C. Greubel, Aug 28 2019

Formula

G.f.: x*(1 +120*x +1191*x^2 +2416*x^3 +1191*x^4 +120*x^5 +x^6)/(1-x)^11.
a(n) = n*(1+n)*(2+n)*(3+n)*(6 -6*n -20*n^2 +15*n^3 +25*n^4 +9*n^5 +n^6)/720.
E.g.f.: x (720 +46440*x +262440*x^2 +425910*x^3 +285264*x^4 +92526*x^5 +15600*x^6 +1380*x^7 +60*x^8 +x^9)*exp(x)/6!. - G. C. Greubel, Aug 28 2019

A255181 Third differences of seventh powers (A001015).

Original entry on oeis.org

1, 125, 1806, 10206, 35406, 92526, 201726, 388206, 682206, 1119006, 1738926, 2587326, 3714606, 5176206, 7032606, 9349326, 12196926, 15651006, 19792206, 24706206, 30483726, 37220526, 45017406, 53980206, 64219806, 75852126, 88998126, 103783806, 120340206
Offset: 0

Views

Author

Luciano Ancora, Mar 18 2015

Keywords

Examples

			Third differences:   1, 125, 1806, 10206, 35406, ...  (this sequence)
Second differences:  1, 126, 1932, 12138, 47544, ...  (A255177)
First differences:   1, 127, 2059, 14197, 61741, ...  (A022523)
---------------------------------------------------------------------
The seventh powers:  1, 128, 2187, 16384, 78125, ...  (A001015)
---------------------------------------------------------------------

Links

Crossrefs

Cf. A000541, A001015, A022523, A255177.

Programs

  • Magma
    [1,125] cat [42*(3-10*n+15*n^2-10*n^3+5*n^4): n in [2..30]]; // Vincenzo Librandi, Mar 18 2015
  • Mathematica
    Join[{1, 125}, Table[42 (3 - 10 n + 15 n^2 - 10 n^3 + 5 n^4), {n, 2, 30}]]

Formula

G.f.: (1 + 120*x + 1191*x^2 + 2416*x^3 + 1191*x^4 + 120*x^5 + x^6)/(1 - x)^5.
a(n) = 42*(3 - 10*n + 15*n^2 - 10*n^3 + 5*n^4) for n>1, a(0)=1, a(1)=125.
a(n) = A255177(n)-A255177(n-1). - R. J. Mathar, Jul 16 2015

Extensions

Edited by Bruno Berselli, Mar 19 2015

A279640 Exponential transform of the seventh powers A001015.

Original entry on oeis.org

1, 1, 129, 2572, 75053, 3228316, 129317797, 6067246144, 321040274585, 17868559819600, 1077981490855241, 69836621117631424, 4779269972787701701, 345917535629669229760, 26385854407691758178669, 2109071203770345994999936, 176264605227324999129677873
Offset: 0

Views

Author

Alois P. Heinz, Dec 16 2016

Keywords

Links

Crossrefs

Column k=7 of A279636.
Cf. A001015.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^7*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);

Formula

E.g.f.: exp(exp(x)*(x^7+21*x^6+140*x^5+350*x^4+301*x^3+63*x^2+x)).

A259907 Fifth differences of 7th powers (A001015).

Original entry on oeis.org

1, 123, 1557, 6719, 16800, 31920, 52080, 77280, 107520, 142800, 183120, 228480, 278880, 334320, 394800, 460320, 530880, 606480, 687120, 772800, 863520, 959280, 1060080, 1165920, 1276800, 1392720, 1513680, 1639680, 1770720, 1906800, 2047920, 2194080, 2345280, 2501520, 2662800
Offset: 0

Views

Author

Kolosov Petro, Jul 07 2015

Keywords

Examples

			1 128 2187 16384 78125 279936 823543 2097152 4782969 (seventh powers)
1 127 2059 14197 61741 201811 543607 1273609 2685817 (first differences)
1 126 1932 12138 47544 140070 341796  730002 1412208 (second differences)
1 125 1806 10206 35406  92526 201726  388206  682206 (third differences)
1 124 1681  8400 25200  57120 109200  186480  294000 (fourth differences)
1 123 1557  6719 16800  31920  52080   77280  107520 (here)

References

  • John H. Conway and Richard K. Guy, The Book of Numbers. New York: Springer-Verlag, pp. 30-32, 1996.
  • Kiran Parulekar. Amazing Properties of Squares and Their Calculations. Kiran Anil Parulekar, 2012.
  • Bag, Amulya Kumar (1966). "Binomial theorem in ancient India". Indian J. History Sci 1 (1): 68-74.
  • Ronald Graham and Donald Knuth, Patashnik, Oren (1994). "(5) Binomial Coefficients". Concrete Mathematics (2nd ed.). Addison Wesley. pp. 153-256.

Links

Crossrefs

Cf. A001015, A022523, A255177, A255181.

Programs

  • Magma
    [1,123,1557,6719] cat [840*(3*n^2-9*n+8): n in [4..40]]; // Bruno Berselli, Jul 16 2015
  • Mathematica
    Join[{1, 123, 1557, 6719}, Table[840 (3 n^2 - 9 n + 8), {n, 4, 40}]]
  • Sage
    [1,123,1557,6719]+[840*(3*n^2-9*n+8) for n in (4..40)] # Bruno Berselli, Jul 16 2015

Formula

G.f.: (1 + 120*x + 1191*x^2 + 2416*x^3 + 1191*x^4 + 120*x^5 + x^6)/(1 - x)^3.
a(n) = 840*(3*n^2 - 9*n + 8) for n>3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>6. - Vincenzo Librandi, Jul 08 2015

Extensions

Edited by Editors of the OEIS, Jul 16 2015

A003369 Numbers that are the sum of 2 positive 7th powers.

Original entry on oeis.org

2, 129, 256, 2188, 2315, 4374, 16385, 16512, 18571, 32768, 78126, 78253, 80312, 94509, 156250, 279937, 280064, 282123, 296320, 358061, 559872, 823544, 823671, 825730, 839927, 901668, 1103479, 1647086, 2097153, 2097280, 2099339, 2113536
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _David A. Corneth_, Aug 03 2020: (Start)
3909794986386 is in the sequence as 3909794986386 = 57^7 + 57^7.
6061605477062 is in the sequence as 6061605477062 = 19^7 + 67^7.
26019535290982 is in the sequence as 26019535290982 = 61^7 + 81^7. (End)

Links

Crossrefs

Cf. A000404 (2 squares), A003325 (2 cubes), A003336 (2 4th), A003347 (2 5th), A003358 (2 6th), A088719 (2 distinct 7th), A003380 (2 8th).
Cf. A001015 (seventh powers).

Programs

  • Maple
    N:= 10^7: # to get all terms <= N
S:= select(`<=`, {seq(seq(a^7+b^7, a=1..b), b=1..floor(N^(1/7)))}, N):
sort(convert(S, list)); # Robert Israel, Sep 03 2017
  • Mathematica
    lst={}; Do[If[(a^7+b^7)==n, AppendTo[lst, n]], {n, 200000}, {a, (n/2)^(1/7)}, {b, a, (n-a^7)^(1/7)}]; lst (* XU Pingya, Sep 03 2017 *)
Module[{upto=10},Select[Union[Total/@Tuples[Range[upto]^7,2]],#<= (upto^7)&]] (* Harvey P. Dale, Feb 04 2019 *)

A003379 Numbers that are the sum of 12 positive 7th powers.

Original entry on oeis.org

12, 139, 266, 393, 520, 647, 774, 901, 1028, 1155, 1282, 1409, 1536, 2198, 2325, 2452, 2579, 2706, 2833, 2960, 3087, 3214, 3341, 3468, 3595, 4384, 4511, 4638, 4765, 4892, 5019, 5146, 5273, 5400, 5527, 5654, 6570, 6697, 6824, 6951, 7078, 7205, 7332, 7459, 7586, 7713
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _David A. Corneth_, Aug 03 2020: (Start)
367634 is in the sequence as 367634 = 3^7 + 4^7 + 4^7 + 4^7 + 4^7 + 4^7 + 4^7 + 4^7 + 4^7 + 5^7 + 5^7 + 5^7.
456545 is in the sequence as 456545 = 2^7 + 2^7 + 2^7 + 4^7 + 4^7 + 4^7 + 4^7 + 5^7 + 5^7 + 5^7 + 5^7 + 5^7.
755708 is in the sequence as 755708 = 1^7 + 2^7 + 2^7 + 3^7 + 3^7 + 3^7 + 4^7 + 4^7 + 5^7 + 5^7 + 6^7 + 6^7. (End)

Links

Crossrefs

Cf. A001015 (seventh powers).
Cf. A003369 - A003378 (numbers that are the sum of 2, ..., 11 positive 7th powers); A003335, A003346, A003357, A003368, A003390, A004801, A004812, A004823 (numbers that are the sum of 12 positive 3rd, ..., 11th powers).

Programs

  • PARI
    (A003379_upto(N, k=12, m=7)=[n|n<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, n)])(8000) \\ 2nd & 3rd optional arg allow to get other sequences of this group. See A003333 for alternate code. - M. F. Hasler, Aug 03 2020

Extensions

Offset corrected by David A. Corneth, Aug 03 2020
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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