A001844 -id:A001844 - cp's OEIS frontend

A254229 Indices of centered square numbers (A001844) which are also heptagonal numbers (A000566).

1, 46, 207, 14652, 66493, 4717738, 21410379, 1519096824, 6894075385, 489144459430, 2219870863431, 157502996839476, 714791523949237, 50715475837851682, 230160650840790723, 16330225716791401968, 74111014779210663409, 5258281965330993581854

Offset: 1

Colin Barker, Jan 27 2015

Also positive integers y in the solutions to 5*x^2 - 4*y^2 - 3*x + 4*y - 2 = 0, the corresponding values of x being A254228.

			46 is in the sequence because the 46th centered square number is 4141, which is also the 41st heptagonal number.

Colin Barker, Table of n, a(n) for n = 1..797
Index entries for linear recurrences with constant coefficients, signature (1,322,-322,-1,1).

Cf. A000566, A001844, A254228, A254230.

LinearRecurrence[{1,322,-322,-1,1},{1,46,207,14652,66493},20] (* Harvey P. Dale, Sep 19 2022 *)

Vec(x*(45*x^3+161*x^2-45*x-1)/((x-1)*(x^2-18*x+1)*(x^2+18*x+1)) + O(x^100))

a(n) = a(n-1)+322*a(n-2)-322*a(n-3)-a(n-4)+a(n-5).

G.f.: x*(45*x^3+161*x^2-45*x-1) / ((x-1)*(x^2-18*x+1)*(x^2+18*x+1)).

A254230 Heptagonal numbers (A000566) which are also centered square numbers (A001844).

Original entry on oeis.org

1, 4141, 85285, 429332905, 8842505113, 44514094237813, 916808615026525, 4615310318335580305, 95056550814337645681, 478524604381155542930941, 9855653300615347164456661, 49614388026831658683830230201, 1021853845419343873890857865865

Offset: 1

Colin Barker, Jan 27 2015

			4141 is in the sequence because it is the 41st heptagonal number and the 46th centered square number.

Colin Barker, Table of n, a(n) for n = 1..399
Index entries for linear recurrences with constant coefficients, signature (1,103682,-103682,-1,1).

Cf. A000566, A001844, A254228, A254229.

Vec(-x*(x^4+4140*x^3-22538*x^2+4140*x+1)/((x-1)*(x^2-322*x+1)*(x^2+322*x+1)) + O(x^100))

a(n) = a(n-1)+103682*a(n-2)-103682*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+4140*x^3-22538*x^2+4140*x+1) / ((x-1)*(x^2-322*x+1)*(x^2+322*x+1)).

A254709 Indices of pentagonal numbers (A000326) which are also centered square numbers (A001844).

Original entry on oeis.org

1, 2, 10, 25, 137, 346, 1906, 4817, 26545, 67090, 369722, 934441, 5149561, 13015082, 71724130, 181276705, 998988257, 2524858786, 13914111466, 35166746297, 193798572265, 489809589370, 2699265900242, 6822167504881, 37595924031121, 95020535478962

Offset: 1

Colin Barker, Feb 06 2015

Also positive integers x in the solutions to 3*x^2 - 4*y^2 - x + 4*y - 2 = 0, the corresponding values of y being A254710.

			10 is in the sequence because the 10th pentagonal number is 145, which is also the 9th centered square number.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,14,-14,-1,1).

Cf. A000326, A001844, A254710, A254711.

LinearRecurrence[{1,14,-14,-1,1},{1,2,10,25,137},30] (* Harvey P. Dale, Aug 08 2017 *)

Vec(-x*(x^4+x^3-6*x^2+x+1)/((x-1)*(x^2-4*x+1)*(x^2+4*x+1)) + O(x^100))

a(n) = a(n-1)+14*a(n-2)-14*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+x^3-6*x^2+x+1) / ((x-1)*(x^2-4*x+1)*(x^2+4*x+1)).

A254710 Indices of centered square numbers (A001844) which are also pentagonal numbers (A000326).

Original entry on oeis.org

1, 2, 9, 22, 119, 300, 1651, 4172, 22989, 58102, 320189, 809250, 4459651, 11271392, 62114919, 156990232, 865149209, 2186591850, 12049974001, 30455295662, 167834486799, 424187547412, 2337632841179, 5908170368100, 32559025289701, 82290197605982

Offset: 1

Colin Barker, Feb 06 2015

Also positive integers y in the solutions to 3*x^2 - 4*y^2 - x + 4*y - 2 = 0, the corresponding values of x being A254709.

			9 is in the sequence because the 9th centered square number is 145, which is also the 10th pentagonal number.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,14,-14,-1,1).

Cf. A000326, A001844, A254709, A254711.

Vec(x*(x^3+7*x^2-x-1)/((x-1)*(x^2-4*x+1)*(x^2+4*x+1)) + O(x^100))

a(n) = a(n-1)+14*a(n-2)-14*a(n-3)-a(n-4)+a(n-5).

G.f.: x*(x^3+7*x^2-x-1) / ((x-1)*(x^2-4*x+1)*(x^2+4*x+1)).

A254711 Pentagonal numbers (A000326) which are also centered square numbers (A001844).

Original entry on oeis.org

1, 5, 145, 925, 28085, 179401, 5448301, 34802825, 1056942265, 6751568605, 205041351065, 1309769506501, 39776965164301, 254088532692545, 7716526200523285, 49291865572847185, 1496966305936352945, 9562367832599661301, 290403746825451948001

Offset: 1

Colin Barker, Feb 06 2015

			145 is in the sequence because it is the 10th pentagonal number and the 9th centered square number.

Colin Barker, Table of n, a(n) for n = 1..875
Index entries for linear recurrences with constant coefficients, signature (1,194,-194,-1,1).

Cf. A000326, A001844, A254709, A254710.

Vec(-x*(x^4+4*x^3-54*x^2+4*x+1)/((x-1)*(x^2-14*x+1)*(x^2+14*x+1)) + O(x^100))

a(n) = a(n-1)+194*a(n-2)-194*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+4*x^3-54*x^2+4*x+1) / ((x-1)*(x^2-14*x+1)*(x^2+14*x+1)).

A254895 Indices of octagonal numbers (A000567) that are also centered square numbers (A001844).

Original entry on oeis.org

1, 13, 53, 1241, 5161, 121573, 505693, 11912881, 49552721, 1167340733, 4855660933, 114387478921, 475805218681, 11208805593493, 46624055769773, 1098348560683361, 4568681660219041, 107626950141375853, 447684178645696213, 10546342765294150201

Offset: 1

Colin Barker, Feb 10 2015

Also positive integers x in the solutions to 6*x^2 - 4*y^2 - 4*x + 4*y - 2 = 0, the corresponding values of y being A253673.

			13 is in the sequence because the 13th octagonal number is 481, which is also the 16th centered square number.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,98,-98,-1,1).

Cf. A000567, A001844, A253673, A254896.

Vec(-x*(x^4+12*x^3-58*x^2+12*x+1)/((x-1)*(x^2-10*x+1)*(x^2+10*x+1)) + O(x^100))

a(n) = a(n-1)+98*a(n-2)-98*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+12*x^3-58*x^2+12*x+1) / ((x-1)*(x^2-10*x+1)*(x^2+10*x+1)).

A254896 Octagonal numbers (A000567) that are also centered square numbers (A001844).

Original entry on oeis.org

1, 481, 8321, 4617761, 79897441, 44339739841, 767175219361, 425750177334721, 7366416376406081, 4088053158428250401, 70732329279075969601, 39253486001477883014881, 679171818371271083701921, 376911968498137474280636161, 6521407729268615666629875041

Offset: 1

Colin Barker, Feb 10 2015

			481 is in the sequence because it is the 13th octagonal number and the 16th centered square number.

Colin Barker, Table of n, a(n) for n = 1..503
Index entries for linear recurrences with constant coefficients, signature (1,9602,-9602,-1,1).

Cf. A000567, A001844, A253673, A254895.

LinearRecurrence[{1,9602,-9602,-1,1},{1,481,8321,4617761,79897441},30] (* Harvey P. Dale, Feb 04 2017 *)

Vec(-x*(x^4+480*x^3-1762*x^2+480*x+1)/((x-1)*(x^2-98*x+1)*(x^2+98*x+1)) + O(x^100))

a(n) = a(n-1)+9602*a(n-2)-9602*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+480*x^3-1762*x^2+480*x+1) / ((x-1)*(x^2-98*x+1)*(x^2+98*x+1)).

A254373 Digital roots of centered square numbers (A001844).

Original entry on oeis.org

1, 5, 4, 7, 5, 7, 4, 5, 1, 1, 5, 4, 7, 5, 7, 4, 5, 1, 1, 5, 4, 7, 5, 7, 4, 5, 1, 1, 5, 4, 7, 5, 7, 4, 5, 1, 1, 5, 4, 7, 5, 7, 4, 5, 1, 1, 5, 4, 7, 5, 7, 4, 5, 1, 1, 5, 4, 7, 5, 7, 4, 5, 1, 1, 5, 4, 7, 5, 7, 4, 5, 1, 1, 5, 4, 7, 5, 7, 4, 5, 1, 1, 5, 4, 7, 5

Offset: 1

Colin Barker, Jan 29 2015

The sequence is periodic with period 9.

			a(3) = 4 because the 3rd centered square number is 13, the digital root of which is 4.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A001844, A010888.

FixedPoint[Plus @@ IntegerDigits[#] &, #] & /@ Table[2 n (n + 1) + 1, {n, 0, 80}] (* Michael De Vlieger, Feb 01 2015 *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 5, 4, 7, 5, 7, 4, 5, 1},86] (* Ray Chandler, Aug 26 2015 *)

PARI
```
m=4; vector(200, n, (m*n*(n-1)/2)%9+1)
```

a(n) = A010888(A001844(n)).
a(n) = a(n-9).
G.f.: -x*(x^8+5*x^7+4*x^6+7*x^5+5*x^4+7*x^3+4*x^2+5*x+1) / ((x-1)*(x^2+x+1)*(x^6+x^3+1)).

A272267 The union of centered square numbers (A001844) and centered 9-gonal numbers (A060544).

Original entry on oeis.org

1, 5, 10, 13, 25, 28, 41, 55, 61, 85, 91, 113, 136, 145, 181, 190, 221, 253, 265, 313, 325, 365, 406, 421, 481, 496, 545, 595, 613, 685, 703, 761, 820, 841, 925, 946, 1013, 1081, 1105, 1201, 1225, 1301, 1378, 1405, 1513, 1540, 1625, 1711, 1741, 1861, 1891

Offset: 1

Colin Barker, Apr 24 2016

nonn

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A001844, A060544, A272266.

cpg(m, n) = m*n*(n-1)/2+1 \\ n-th centered m-gonal number
cpgr(m, r) = n=1; L=List(); while((t=cpg(m, n))

Conjectures:
  a(n) = a(n-1)+2*a(n-5)-2*a(n-6)-a(n-10)+a(n-11) for n>11.
  G.f.: x*(1+4*x+5*x^2+3*x^3+12*x^4+x^5+5*x^6+4*x^7+x^10) / ((1-x)^3*(1+x+x^2+x^3+x^4)^2).

A278244 Least number with the prime signature of the n-th centered square number (A001844(n)).

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 6, 2, 6, 2, 6, 6, 2, 6, 2, 6, 6, 2, 6, 2, 4, 12, 2, 30, 2, 2, 6, 6, 24, 2, 2, 6, 2, 6, 2, 2, 30, 6, 6, 2, 6, 30, 2, 6, 6, 6, 12, 2, 6, 12, 2, 6, 6, 12, 6, 6, 6, 6, 12, 6, 2, 30, 6, 6, 6, 2, 30, 6, 6, 2, 2, 12, 2, 6, 6, 6, 6, 6, 60, 2, 6, 6, 2, 6, 2, 2, 30, 2, 30, 6, 2, 30, 6, 30, 6, 30, 24, 2, 6, 2, 2, 30, 2, 12, 2, 6, 30, 6, 30, 2, 2, 6, 6, 6, 6

Offset: 0

Antti Karttunen, Nov 22 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A001844, A046523.

Cf. also A278162, A278254, A278259.

A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From Charles R Greathouse IV, Aug 17 2011
A001844(n) = 2*((n^2) + n) + 1;
A278244(n) = A046523(A001844(n));
for(n=0, 10000, write("b278244.txt", n, " ", A278244(n)));

(define (A278244 n) (A046523 (A001844 n)))

a(n) = A046523(A001844(n)).

cp's OEIS Frontend

A254229 Indices of centered square numbers (A001844) which are also heptagonal numbers (A000566).

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A254230 Heptagonal numbers (A000566) which are also centered square numbers (A001844).

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A254709 Indices of pentagonal numbers (A000326) which are also centered square numbers (A001844).

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A254710 Indices of centered square numbers (A001844) which are also pentagonal numbers (A000326).

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A254711 Pentagonal numbers (A000326) which are also centered square numbers (A001844).

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A254895 Indices of octagonal numbers (A000567) that are also centered square numbers (A001844).

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A254896 Octagonal numbers (A000567) that are also centered square numbers (A001844).

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A254373 Digital roots of centered square numbers (A001844).

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