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 1-9 of 9 results.

A292496 Number of solutions to +- 1^2 +- 3^2 +- 5^2 +- 7^2 +- ... +- (4*n-1)^2 = 0.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 12, 0, 40, 10, 516, 124, 5020, 1828, 48570, 32806, 527890, 444480, 6137942, 6482314, 70573856, 93276044, 853480374, 1300190254, 10660384742, 18371629260, 134129890382, 259804151324, 1728886287134, 3667061002286, 22672130669968
Offset: 0

Views

Author

Seiichi Manyama, Sep 17 2017

Keywords

Examples

			For n=4 the 2 solutions are +1^2-3^2-5^2+7^2-9^2+11^2+13^2-15^2 = 0 and -1^2+3^2+5^2-7^2+9^2-11^2-13^2+15^2 = 0.

Links

Crossrefs

Cf. A158092, A292476, A292497.

Programs

  • PARI
    a(n) = polcoeff(prod(k=1, 2*n, x^(2*k-1)^2+1/x^(2*k-1)^2), 0); \\ Michel Marcus, Sep 18 2017

Formula

Constant term in the expansion of Product_{k=1..2*n} (x^(2*k-1)^2+1/x^(2*k-1)^2).

A006718 Number of golygons of length 8n.

Original entry on oeis.org

1, 4, 112, 8432, 909288, 121106960, 18167084064, 2956370702688, 510696155882492, 92343039606440064, 17311893232788414400, 3342127071364266721200, 661066887819006986788620, 133456726466163517072371360
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

A007219 is the main entry for golygons.

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 92.

Links

Crossrefs

Cf. A060468, A063865, A292476.
See A007219 for much more information about golygons.

Programs

  • Mathematica
    p1[n_] := Product[x^k + 1, {k, 1, n - 1, 2}] // Expand; p2[n_] := Product[x^k + 1, {k, 1, n/2}] // Expand; c[n_] := Coefficient[p1[n], x, n^2/8] * Coefficient[p2[n], x, n (n/2 + 1)/8]; a[n_] := c[8*n]; Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Jul 24 2013, after Eric W. Weisstein *)

Formula

a(n) = 4 * A007219(n) for n > 0. - Charles R Greathouse IV, Apr 29 2012
a(n) = A060468(n) * A292476(2*n) = A063865(4*n) * A292476(2*n). - Seiichi Manyama, Sep 18 2017

Extensions

a(0) = 1 prepended by Seiichi Manyama, Sep 18 2017

A107350 Number of isogons with a certain property.

Original entry on oeis.org

1, 4, 34, 346, 3965, 48396, 615966, 8082457, 108545916, 1484716135, 20612084010, 289688970195, 4113620233260, 58930127470164, 850641610106596, 12360278974175769, 180648953113093368, 2653875476976308643, 39167191622334514398, 580439539153823110678, 8633956582855204662785
Offset: 1

Views

Author

N. J. A. Sloane, May 23 2005

Keywords

Comments

This and A060005 appear in the reference as incidental sequences when computing A007219.

Links

Crossrefs

Cf. A007219, A060005, A292476.

Programs

  • Maple
    A107350 := proc(n) res := 1 ; for i from 0 to 4*n-1 do res := taylor(res*(1+x^(2*i+1)),x=0,8*n^2+1) ; od ; coeftayl(res,x=0,8*n^2)/2 ; end: for n from 1 to 25 do printf("%d, ",A107350(n)) ; od ; # R. J. Mathar, May 08 2007
  • Mathematica
    a[n_] := SeriesCoefficient[Product[x^(2k - 1) + 1/x^(2k - 1), {k, 1, 4n}], {x, 0, 0}]/2;
Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Mar 10 2023 *)

Formula

product[ {1+x^(2i+1)},i=0,1,...,4n-1] = 1+...+2*a(n)*x^(8n^2)+.... (g.f.). - R. J. Mathar, May 08 2007
a(n) = A292476(2*n)/2. - Seiichi Manyama, Sep 18 2017

Extensions

More terms from R. J. Mathar, May 08 2007

A369343 a(n) is the constant term in expansion of Product_{k=1..n} (x^(2*k-1) + 1 + 1/x^(2*k-1)).

Original entry on oeis.org

1, 1, 1, 1, 3, 9, 21, 49, 117, 295, 761, 1993, 5261, 14025, 37699, 102151, 278587, 764145, 2106433, 5832863, 16217191, 45255167, 126708863, 355848715, 1002145705, 2829479797, 8007670701, 22711890561, 64547494347, 183790615881, 524239904367, 1497786769295
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 20 2024

Keywords

Comments

All terms are odd.

Links

Crossrefs

Cf. A005408, A007576, A156700, A292476.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n>i^2, 0, `if`(i=0, 1,
      b(n, i-1)+b(n+2*i-1, i-1)+b(abs(n-2*i+1), i-1)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 21 2024
  • Mathematica
    Table[Coefficient[Product[x^(2 k - 1) + 1 + 1/x^(2 k - 1), {k, 1, n}], x, 0], {n, 0, 30}]

Formula

a(n) ~ 3^(n+1) / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jan 21 2024

A292522 Number of solutions to +- 1^3 +- 3^3 +- 5^3 +- 7^3 +- ... +- (4*n-1)^3 = 0.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 2, 6, 2, 10, 118, 88, 254, 3308, 2558, 9578, 84568, 121804, 496396, 3312400, 5755724, 19021024, 116780256, 241754350, 883730786, 4923089216, 11668601596, 42357336066, 205859270250, 538878582526, 1974181071852, 9194146886086, 26277093562150
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2017

Keywords

Examples

			For n=8 the 2 solutions are
+1^3-3^3-5^3+7^3-9^3+11^3+13^3-15^3-17^3+19^3+21^3-23^3+25^3-27^3-29^3+31^3 = 0 and
-1^3+3^3+5^3-7^3+9^3-11^3-13^3+15^3+17^3-19^3-21^3+23^3-25^3+27^3+29^3-31^3 = 0.

Crossrefs

Cf. A158118, A292476, A292496.

Formula

Constant term in the expansion of Product_{k=1..2*n} (x^((2*k-1)^3)+1/x^((2*k-1)^3)).

Extensions

a(29)-a(34) from Alois P. Heinz, Sep 18 2017

A367087 Number of solutions to +- 1 +- 3 +- 5 +- 7 +- ... +- (2*n-1) = 0 or 1.

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 2, 5, 8, 13, 20, 38, 68, 118, 206, 380, 692, 1262, 2306, 4277, 7930, 14745, 27492, 51541, 96792, 182182, 343670, 650095, 1231932, 2338706, 4447510, 8472697, 16164914, 30884150, 59086618, 113189168, 217091832, 416839177, 801247614, 1541726967, 2969432270
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 26 2024

Keywords

Crossrefs

Cf. A005408, A025591, A156700, A292476.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n>i^2, 0,
      `if`(i=0, 1, b(n+2*i-1, i-1)+b(abs(n-2*i+1), i-1)))
    end:
a:=n-> b(irem(n, 2), n):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 26 2024
  • Mathematica
    b[n_, i_] := b[n, i] = If[n > i^2, 0,
   If[i == 0, 1, b[n+2*i-1, i-1] + b[Abs[n-2*i+1], i-1]]];
a[n_] := b[Mod[n, 2], n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 03 2025, after Alois P. Heinz *)

A292550 a(n) = smallest k >= 1 such that {1, 3^n, 5^n, ... , (4*k-1)^n} can be partitioned into two sets with equal sums.

Original entry on oeis.org

1, 2, 4, 8, 10, 14, 19
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2017

Keywords

Examples

			n = 0
1^0 = 3^0.
n = 1
1^1 + 7^1 = 3^1 + 5^1.
n = 2
1^2 + 7^2 + 11^2 + 13^2 = 3^2 + 5^2 + 9^2 + 15^2.
n = 3
1^3 + 7^3 + 11^3 + 13^3 + 19^3 + 21^3 + 25^3 + 31^3 = 3^3 + 5^3 + 9^3 + 15^3 + 17^3 + 23^3 + 27^3 + 29^3.
n = 4
1^4 + 5^4 + 13^4 + 17^4 + 19^4 + 25^4 + 27^4 + 29^4 + 31^4 + 39^4 = 3^4 + 7^4 + 9^4 + 11^4 + 15^4 + 21^4 + 23^4 + 33^4 + 35^4 + 37^4.
n = 5
1^5 + 3^5 + 7^5 + 11^5 + 17^5 + 21^5 + 33^5 + 35^5 + 37^5 + 39^5 + 41^5 + 43^5 + 51^5 + 53^5 = 5^5 + 9^5 + 13^5 + 15^5 + 19^5 + 23^5 + 25^5 + 27^5 + 29^5 + 31^5 + 45^5 + 47^5 + 49^5 + 55^5.

Crossrefs

Cf. A019568 (similar sequence).
Cf. A292476, A292496, A292522.

Extensions

a(5)-a(6) from Alois P. Heinz, Sep 18 2017

A369386 a(n) is the constant term in expansion of Product_{k=1..n} (x^(2*k-1) + 1/x^(2*k-1))^2.

Original entry on oeis.org

1, 2, 4, 8, 18, 48, 138, 428, 1392, 4652, 15884, 55124, 193724, 688008, 2465134, 8899700, 32342236, 118215780, 434314138, 1602935104, 5940303754, 22095769648, 82464791420, 308715131744, 1158949678600, 4362040367048, 16456820491806, 62223707844096
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2024

Keywords

Crossrefs

Cf. A047653, A156181, A156700, A292476.

Programs

  • Mathematica
    Table[Coefficient[Product[(x^(2 k - 1) + 1/x^(2 k - 1))^2, {k, 1, n}], x, 0], {n, 0, 27}]

A369729 Number of solutions to +- 1 +- 3 +- 5 +- 7 +- ... +- (4*n-3) = 1.

Original entry on oeis.org

0, 1, 1, 2, 5, 13, 38, 118, 380, 1262, 4277, 14745, 51541, 182182, 650095, 2338706, 8472697, 30884150, 113189168, 416839177, 1541726967, 5724470097, 21330062502, 79733319862, 298922247363, 1123678419818, 4234465089737, 15993581636893, 60535561889465
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2024

Keywords

Crossrefs

Cf. A005408, A063866, A156700, A292476, A367087.

Programs

  • Mathematica
    Table[Coefficient[Product[(x^(2 k - 1) + 1/x^(2 k - 1)), {k, 1, 2 n - 1}], x, 1], {n, 0, 27}]

Formula

a(n) = [x^1] Product_{k=1..2*n-1} (x^(2*k-1) + 1/x^(2*k-1)).
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.